Randomness, Variability, and Sample Size are important, interrelated concepts that help us better understand and predict events in basketball. Randomness (i.e., luck) is a part of basketball–wide-open jumpers will rattle in and out while contested half-court shots bank in off the backboard. Players suffer tragic injuries and refs make bad calls. Among other reasons, this randomness explains why a 40% 3-point shooter will shoot 2-10 one night and 6-10 another night. This disparity in results is called variability. If Ray Allen shoots 3-6 FTs, it’s likely a random fluke not worth worrying about. If he does this for a month straight, then maybe we should start asking questions, which takes us to this section’s final topic, sample size. On a certain level, we all know that it is more impressive for a player to average 50 points per game than it is to score 50 in a single night. Nonetheless, as humans we have a natural tendency to overreact to small samples (which are vulnerable to randomness) and underestimate the power of large samples. Some of the issues most near and dear to our hearts–e.g., the “hot hand,” upsets, and clutch play–are all better understood by acknowledging the importance of sample size and randomness. Below you’ll find brief explanations of how the above phenomena relate to other basketball concepts.

The “Hot Hand” is the belief that a player is more likely to make a shot after having just made one, i.e., that he gets “hot” during a run of made shots. The truth is that streaks can be the product of random events (Example: Until 2012, the NFC had won 14 straight Super Bowl coin-flips). Larger, more frequently occurring streaks would confirm the hot hand’s existence, but such streaks have generally been non-existent in large data sets. Thus, the hot hand is an example of humans reading too much into small samples while ignoring randomness and variability. The power of a large sample is equally as important. If Ray Allen is 5-8 from the free throw line while KG is 10-10, I would still prefer Allen to take a technical free throw. Why? He and Garnett’s larger sample of career free throws show that Allen is the better free throw shooter.

Upsetsare when an underdog beats out a favorite; sometimes they are the result of the the underdog’s “hunger” or the favorite “choking,” but given that a 7-game series is a tiny sample, randomness also plays a part in the result. Consider the following quote from The Drunkard’s Walk: How Randomness Rules Our Lives by Leonard Mlodinow:

“…If the superior team could beat its opponent, on average, 2 out of 3 times they meet, the inferior team will still win a 7-game series about once every 5 match-ups…You’d have to play a series consisting of at minimum the best of 23 games to determine the winner with what is called statistical significance, meaning the weaker team would be crowned champion 5 percent or less of the time. And in the case of one team’s having only a 55-45 edge, the shortest significant “world series” would be the best of 269 games…” (p. 70-71)

Another quick point regarding upsets and variability: underdogs can employ strategies with “high variability” to increase their chances of upsetting a favorite. One such strategy is to shoot 3-pointers, which is where the phrase “Live by the 3, die by the 3” comes from. Think of it this way: if on average the Wizards shoot 33.3 3pt% and 50% 2pt%, they will generally score 100 points per 100 shots. But if I need to beat the Spurs, who score 110 points per 100 shots without fail, I will need to turn to the high-risk, high-reward strategy (i.e., 3-point shooting) to have a good chance at outscoring the Spurs. Likewise, as an underdog, if I have the option between playing the Spurs, who score exactly 110 each game, and the Lakers, who also average 110 ppg but have scores that range between 90 and 130, I’d rather play the Lakers with the hope that in a 7 game series I can catch them during their off-games.

“Clutch play” indicates the belief that some people are inherently better than others in the final seconds of games; in reality, most research has indicated that notable end-game play–be it good or bad–is often the result of randomness. We often witness a player hit a game-winning shot (e.g., Kobe) or falter in the 4th quarter (e.g., LeBron), form impressions based on these brief samples, and let these small samples bias our beliefs and future observations. In reality, larger samples of clutch situations indicate there are no systematic differences for players or teams in clutch play across seasons.

Plus-minus is a powerful concept (what is better than a stat that inherently measures impact on team success?), but it also requires large sample sizes to have any statistical power. Plus-minus falls prey to high variability and multicollinearity, meaning that large, diverse samples must be collected to reduce the errors that come with plus-minus. In other words, plus-minus is unreliable over short samples–a given player’s plus-minus is dependent on his opponents’ and teammates’ performance, which interfere with measuring a player’s “true” impact on team performance. The other murky issue is that players frequently share the floor with the same players time and time again; if I always play alongside Wade, James, and Bosh, then my plus-minus will flourish; if I always share the court with Nick Young, Andray Blatche, and Rashard Lewis, I will probably have a bad plus-minus. Adjusted Plus-Minus and Regularized APM try to account for quality of teammates and opponents, but they also require large samples from various lineups to make the adjustments for teammate/opponents quality. The big picture is this: single-game plus-minus stats mean almost nothing, single-season plus-minus stats mean little, and over time, with the proper adjustments (e.g., RAPM) we can get a clear picture of a player’s impact via plus-minus.

The power of point differentialis a good illustration of the power of sample size. Basketball games are made up of individual possessions, and tracking the outcome of each possession (i.e., scoring or allowing points) provides a larger sample size than does W-L record, which by nature is limited to 82 binary data points for a season sample. When close games (which are largely random) and definitive 30-point margins are treated equally in the standings, you weaken the meaning and power of your data. Point differential carries with it more information and thus is a better predictor of future outcomes (including playoff and championship success) than traditional W-L record.

Tradeoff between usage and efficiencyis the trend that a player’s offensive efficiency will decrease as his load becomes greater (and vice versa). Understanding this tradeoff speaks to the importance of skill curves and player roles when observing a player’s statistics.Certain players, such as spot-up 3-point shooters (e.g., Steve Kerr) and big men that that are set up for easy shots (e.g., Tyson Chandler) will post amazing Offensive Ratings, but these are usually coupled with low Usage Rates. It takes a special player (e.g., Dirk Nowitzki,Lebron James) to score efficiently while using a lot of plays. It might be surprising that Kerr and Grant both have higher career Offensive Ratings than James and Nowitzki, but once we understand what ORtg measures and its dependency on Usg % and Skill Curves, we get a better idea of how to evaluate players in their given roles. If a player has a a high Usg% and a low ORtg, perhaps he should relinquish some of his scoring duties. Likewise, coaches must evaluate a player’s skill set before throwing him into a high-usage role. Trevor Ariza is the poster-boy for the usage-efficiency tradeoff. Observe the below table:

As you can see, Ariza’s scoring efficiency took a drastic dip as his usage increased. The case of Ariza may be extreme, but the principle remains true in general and is important to keep in mind when evaluating players.

Correlation doesn’t imply causation is a classic concept in stats that can be applied to basketball. A correlation is a relationship between to variables, such that as we observe an increase in X, we see an increase (or decrease) in Y. X might be causing a change in Y, Y might be causing a change in X, or some third variable Z might be causing a change in both. In some cases, the direction of cause is ambiguous. Take the observation that “The more shots Andray Blatche misses, the more we hear boos.” Blatche’s misses might be causing the crowd to boo, but it could be the other way around–the crowd’s boos might cause Andray to miss. We cannot know for sure. An example with a third variable is how we might notice that, on average, the more rebounds a player gets per game, the more points he scores per game. Are his points causing rebounds? Certainly not. Are his rebounds causing points? Perhaps, but the best explanation is that a third variable–increased minutes–is causing both to increase. A more nuanced example might be Gregg Popovich and his San Antonio Spurs. We might notice that in seasons where the Spurs win championships, the Spurs starters play fewer minutes. From this, some will gather that because Popovich is sitting his stars more, they are better rested for the playoffs and thus win titles. This could be true, there is another explanation–perhaps we find that Popovich rests his stars more in championship seasons because his teams are better those years and thus find themselves in more blowouts, allowing him the luxury to rest players. Here, we would see a third variable, i.e., the Spurs’ ability, causing both the decreased playing time and the championships, rather than the decreased playing time causing the championships.

Law of diminishing returns states that as you gain more of a certain product, the positive effects from this product will plateau. To make this more clear, let’s take the example of rebounding. Pretend the Wizards are the worst rebounding team and the Bulls are the best rebounding team, and that both Nick Young and Richard Hamilton are putting up identical rebounding numbers for their teams. If we subbed Evan Turner in for both teams (the league’s best rebounding SG), the Wizard’s rebounding would benefit more, as the Bulls are already grabbing a high percentage of available rebounds. Diminishing returns has been shown to exist for defensive rebounding, but not offensive rebounding; in other words players do not take offensive rebounds away from each other. Another example of diminishing returns might be the added value of ball-handling. A team of forwards and centers would benefit greatly from a ball-handler to bring the ball up the floor, but a team playing with four guards likely doesn’t need its other player to be a great ball-handler.